Fourier Series

PERIODIC FUNCTIONS

We say that a function is periodic over an interval with a period T if , for every , implies .

The sum of several periodic functions with periods that are rational numbers is again a periodic function such as

TRIGONOMETRIC SERIES

The question is whether a periodic function can always be expressed as the sum of a finite number of periodic functions or as an infinite series, that is as the sum of an infinite sequence of periodic functions. The answer is that, for certain, sufficiently "decent" functions this can be done using infinite series.

As periodic functions we will use sines and cosines with various multiples of arguments.

Let us assume that a function periodic in can be written as an infinite series

Note that this can also be rewritten in the form

We will try to calculate the coefficients

ORTHOGONAL FUNCTIONS

The functions

are orthogonal in , which means that the integral of any pair of different functions over this interval equals zero whereas the integral of the square of any function over the interval is non-zero.

Indeed, using the well known trigonometric formulas,

it can be established that

and

EULER FOURIER METHOD

The above-demonstrated property of trigonometric functions will help us in establishing the coefficients of the Fourier expansion.

This gives us the formulas for the cosine coefficients

In a similar way we can obtain

and

At one stage of establishing these coefficients, we replaced the integral of an infinite series of functions by an infinite series of the functions integrated separately. This is only possible if the series converges uniformly

POINTWISE AND UNIFORM CONVERGENCE

If is an infinite series the terms

u0(x), u1(x), u2(x), . . . , un(x), . . .

of which are functions defined in an interval of the real variable x and if converges at each point of the interval, then the series is said to converge

pointwise in the interval; f(x) denotes the sum, and Sn(x) the partial sums, of the series.

Pointwise convergence means that, for any positive , there is an index n0, depending in general on and x, in symbols n0 = n0( , x), such that f(x) - Sn(x)< forn > n0.

If, for each > 0, an n0 can be found independent of x, the series is said to converge uniformly in the interval considered.

Geometrically this means that, for each n > n0, the graph of the curve y = sn(x) and that of the curvey = f(x) differ by less than in the whole interval under consideration.

Pointwise convergence in general does not preserve properties of the terms of the series. For example, if the functions un(x) are continuous and converges

pointwise, the sum need not be continuous.

A simple example is provided by the series

(1 - x) + x(1 - x) + x2(1 - x) + … ,

which converges at each point of the interval 0 x 1 and the sum of which is equal to 0 for x = 1 and equal to 1 at the remaining points of the interval.

The significance of the notion of uniform convergence is that the sum of a uniformly convergent series of functions inherits many important properties of the terms – in particular their continuity and integrability.

Theorem

If the un(x) are continuous in a finite interval (a, b) and if converges uniformly

in (a, b), then the sum f(x) of is also continuous in (a, b).

Theorem

If the un(x) are integrable over a finite interval (a, b) and if converges uniformly in (a, b), then the sum f(x) of is integrable over (a, b) and a definite integral of the function f is written as a sum of integrals of functions un.

DIRICHLET CONDITIONS

We say that a function satisfies the Dirichlet conditions in the interval if

1. it is uniformly bounded in , that is, where is a constant,2. it has at most a finite number of points of discontinuity all of which being of the

first type, that is, at each such point of discontinuity , has a finite left limit and a finite right limit where ,

3. it has at most a finite number of proper minima and maxima.

DIRICHLET THEOREM

If a function satisfies the Dirichlet conditions in , at each point in where is continuous, it can be expanded into a Fourier series

where the Fourier coefficients are given by the following formulas

(1)

(2)

If is a point of discontinuity of , then

and, for and , we have

DIRICHLET THEOREM FOR

If a function satisfies the Dirichlet conditions in where , at each point in where is continuous, it can be expanded into a Fourier series

where the Fourier coefficients are given by the following formulas

(3)

(4)

If is a point of discontinuity of , then

and, for and , we have

FOURIER COEFFICIENTSFOR ODD AND EVEN FUNCTIONS

If a function is odd, that is, , then the formulas (1) and (2) will take the following form

(1ODD)

(2ODD).

Similarly, for an even function , that is, , formulas (3) and (4) can be rewritten as

(1EVEN)

(2EVEN)

In an analogous way, we can derive formulas (3ODD), (4ODD) and (3EVEN), (4EVEN).

EVEN, ODD EXTENSION OF A FUNCTION

Let be a function given in the interval that satisfies the Dirichlet conditions and let . Let us define a function in the interval as follows

if , if ,

for .

Similarly, we can define in a function

if , if ,

for .

FOURIER EXPANSION IN THE INTERVAL

Both and satisfy the Dirichlet conditions and so can be expanded into Fourier series. Since, clearly, is odd and is even, we can use the formulas (3ODD), (4ODD) and (3EVEN), (4EVEN) respectively.

Since the expansions can then be restricted to the interval , we can view this process as an expansion of the original function in the interval .

EXAMPLE 1

Find the Fourier expansion for the function

SOLUTION

EXAMPLE 2

Find the Fourier expansion for the function in .

SOLUTION

EXAMPLE 3

Find the Fourier expansion for the function in and, using the result, find the sum of the Cauchy series

SOLUTION

Substituting in the above formula, we obtain

EXAMPLE 4

Find the Fourier expansion for the function where c is a constant.

SOLUTION

We can either define or . For , obviously and, for ,

we obtain

Thus for, say, we have

Fourier Series

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